On the spectrum of the Page and the Chen–LeBrun–Weber metrics
We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen–LeBrun–Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the estimat...
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| Published in: | Annals of global analysis and geometry Vol. 46; no. 1; pp. 87 - 101 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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01.06.2014
Springer Nature B.V |
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| ISSN: | 0232-704X, 1572-9060 |
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| Abstract | We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen–LeBrun–Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the estimation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue. |
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| AbstractList | We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the estimation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue. [PUBLICATION ABSTRACT] We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen–LeBrun–Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the estimation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue. |
| Author | Murphy, Thomas Hall, Stuart J. |
| Author_xml | – sequence: 1 givenname: Stuart J. surname: Hall fullname: Hall, Stuart J. email: stuart.hall@buckingham.ac.uk organization: Department of Applied Computing, University of Buckingham – sequence: 2 givenname: Thomas surname: Murphy fullname: Murphy, Thomas organization: Départment de Mathématique, Université Libre de Bruxelles, Department of Mathematics, McMaster University |
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| Cites_doi | 10.1007/s12220-012-9343-z 10.1112/plms/84.1.213 10.1142/S0129167X98000282 10.4310/PAMQ.2009.v5.n2.a2 10.1007/s00220-008-0558-6 10.1090/S0894-0347-08-00594-8 10.1103/PhysRevD.28.2436 10.1007/s00208-011-0701-0 10.1017/S0027763000014847 10.1201/9781003072393-11 10.1088/1126-6708/2008/07/120 |
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| DOI | 10.1007/s10455-014-9412-6 |
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| Snippet | We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen–LeBrun–Weber Einstein metrics. One notable feature is... We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is... |
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| SubjectTerms | Analysis Approximation Differential Geometry Eigenvalues Explicit knowledge Geometry Global Analysis and Analysis on Manifolds Laplace transforms Mathematical models Mathematical Physics Mathematics Mathematics and Statistics Studies Theorems |
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| Title | On the spectrum of the Page and the Chen–LeBrun–Weber metrics |
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